Parametric constraints for Bayesian knowledge tracing from first principles

Bayesian Knowledge Tracing (BKT) is a probabilistic model of a learner’s state of mastery for a knowledge component. The learner’s state is a “hidden” binary variable updated based on the correctness of the learner’s responses to questions corresponding to that knowledge component. The parameters used for this update are inferred/learned from historical ground truth data. For this, BKT is often represented as a Hidden Markov Model and the Expectation-Maximization algorithm is used to infer the parameters. The algorithm, can however, suffer from issues including settling into local minima, producing degenerate parameter values (such as stating that learners who do not know the skill are more likely to answer correctly than those who do), and a high computational cost during fitting. To address these, we take a “from first principles” approach to derive necessary constraints that can be imposed on the BKT parameter space. Starting from the basic mathematical truths of prob-ability and using conceptual behaviors expected of the BKT parameters in real systems, we derive succinct constraints for the BKT parameter space. As necessary conditions, applying the constraints prior to fitting reduces computational cost and the issues emerging from the EM procedure. We further introduce a novel algorithm for estimating BKT parameters subject to the newly defined constraints. While the issue of degenerate parameters has been reported previously, this paper is the first, to the best of our knowledge, to derive necessary constraints from first principles and also present an algorithm that respects those constraints.